Kurdyka- Lojasiewicz inequality and Kurdyka- Lojasiewicz - - PowerPoint PPT Presentation

Kurdyka-

• Lojasiewicz

inequality and subgradient trajectories: the convex case Olivier Ley Journ´ ees Franco- Chiliennes Toulon, April 2008

Kurdyka- Lojasiewicz inequality and subgradient trajectories : the convex case

Olivier Ley Universit´ e de Tours www.lmpt.univ-tours.fr/∼ley Joint work with : J´ erˆ

• me Bolte (Paris vi)

Aris Daniilidis (U. Autonoma Barcelona & Tours) and Laurent Mazet (Paris xii)

Kurdyka-

• Lojasiewicz

inequality and subgradient trajectories: the convex case Olivier Ley Journ´ ees Franco- Chiliennes Toulon, April 2008

• Lojasiewicz inequality
• Lojasiewicz inequality [

Lojasiewicz 1963]

f : RN → R is analytic. Let a be a critical point of f . Then there exists a neighborhood U of a, C > 0 and θ ∈ (0, 1) such that ||∇f (x)|| ≥ |f (x) − f (a)|θ for all x ∈ U. ➪ Finite length of the gradient trajectories, Every critical point is limit of a gradient trajectory, etc.

Kurdyka-

• Lojasiewicz

inequality and subgradient trajectories: the convex case Olivier Ley Journ´ ees Franco- Chiliennes Toulon, April 2008

Basic assumptions

To simplify :

H is a real Hilbert space f : H → [0, +∞) is smooth (f ≥ 0) For all r > 0, Cr := {f ≤ r} (A0) (0 is a critical point and a global minimum) 0 ∈ C0 (A1) (0 is an isolated critical value) There exits r0 > 0 such that : x ∈ Cr0 and f (x) > 0 ⇒ ∇f (x) = 0 (A2) (Sublevel compactness) There exits r0 > such that : Cr0 = {f ≤ r0} is compact.

Kurdyka-

• Lojasiewicz

inequality and subgradient trajectories: the convex case Olivier Ley Journ´ ees Franco- Chiliennes Toulon, April 2008

Generalization : K L-inequality

We say that f satisfies Kurdyka- Lojasiewicz inequality [Kurdyka 1998] if : There exists ϕ ∈ KL(0, r0) such that : ||∇(ϕ ◦ f )(x)|| ≥ 1 for all x ∈ Cr0 \ C0. where : KL(0, r0) =

• ϕ : [0, r0] → R+ continuous,

ϕ(0) = 0, ϕ ∈ C 1(0, r0), ϕ′ > 0

• .

◮ Lojasiewicz inequality is a particular case with ϕ(r) =

1 C(1−θ)r1−θ.

Kurdyka-

• Lojasiewicz

inequality and subgradient trajectories: the convex case Olivier Ley Journ´ ees Franco- Chiliennes Toulon, April 2008

The convex case

From now on, we assume that f is convex Issues :

1 Characterizations of the K

L-inequality in the convex case

2 Does a convex function satisfy the K

L-inequality ?

Kurdyka-

• Lojasiewicz

inequality and subgradient trajectories: the convex case Olivier Ley Journ´ ees Franco- Chiliennes Toulon, April 2008

Piecewise gradient curves

Gradient dynamical system : ˙ Xx(t) = −∇f (Xx(t)), t ≥ 0 Xx(0) = x A piecewise gradient curve γ is a countable family of gradient curves Xxi([0, ti)) with f (Xxi(0)) = f (xi) = ri, f (Xxi(ti)) = ri+1 ri ↓

i→+∞

Cr0 Cr2 Cr1 Cr3 C0

x0 Xx0(t) x1 x2

Kurdyka-

• Lojasiewicz

inequality and subgradient trajectories: the convex case Olivier Ley Journ´ ees Franco- Chiliennes Toulon, April 2008

Classical properties of the ‘convex’ gradient curves

• Lemma. For all x0 ∈ Cr0 \ C0,

1 t → f (Xx0(t)) is convex, L1(0, +∞) and decreasing with

limit 0.

2 Each trajectory goes closer to all minima at the same time,

i.e., for each a ∈ C0, d dt ||Xx0(t) − a||2 ≤ −2f (Xx0(t)) < 0.

3 For all T > 0,

T || ˙ Xx0(t)||dt ≤ 1 √ 2 ||x0||

• log T.

Kurdyka-

• Lojasiewicz

inequality and subgradient trajectories: the convex case Olivier Ley Journ´ ees Franco- Chiliennes Toulon, April 2008

Characterizations of the K L-inequality (f convex)

• Theorem. The following statements are equivalent :

1 f satisfies the K

L-inequality in Cr0 : ||∇(ϕ ◦ f )(x)|| ≥ 1 with ϕ ∈ KL(0, r0).

2 f satisfies the K

L-inequality globally in H with ϕ ∈ KL(0, +∞) which is concave.

3 r ∈ (0, r0] → 1

inf

f (x)=r ||∇f (x)|| is integrable. 4 For all piecewise gradient curves γ in Cr0 we have

length(γ) =

∞

• i=0

ti || ˙ Xxi(t)||dt < ∞.

Kurdyka-

• Lojasiewicz

inequality and subgradient trajectories: the convex case Olivier Ley Journ´ ees Franco- Chiliennes Toulon, April 2008

Length of the ‘convex’ gradient curves

[Br´ ezis 1973] Given x0 ∈ Cr0, do we have length(Xx0) = ∞ || ˙ Xx0(t)||dt < ∞ ?

• Theorem. [Br´

ezis 1973] Yes if int(argmin(f ))= ∅.

C0

finite length finite length unknown length

C0

Xx0(t) ˙ Xx0(t)

• Theorem. [Baillon 1978] No in general (counter-example in

infinite dimension)

Kurdyka-

• Lojasiewicz

inequality and subgradient trajectories: the convex case Olivier Ley Journ´ ees Franco- Chiliennes Toulon, April 2008

A sufficient condition for a convex function to satisfy K L-inequality

• Theorem. Assume that there exists

m : [0, +∞) → [0, +∞) continuous increasing with m(0) = 0 such that f ≥ m(dist(·, C0)) on Cr0 and r0 m−1(r) r dr < +∞ (growth condition). (1) Then K L-inequality holds for f . ◮ Proof : f (x) ≤ ∇f (x), x − pC0(x) ≤ ||∇f (x)||dist(x, C0) ≤ ||∇f (x)||m−1(f (x)). ◮ non analytic examples : m(r) = exp(−1/rα), α ∈ (0, 1) satisfies (1).

Kurdyka-

• Lojasiewicz

inequality and subgradient trajectories: the convex case Olivier Ley Journ´ ees Franco- Chiliennes Toulon, April 2008

A smooth convex counterexample to K L-inequality

• Theorem. There exists a C 2 convex function f : R2 → R+

with {f = 0} = D(0, 1) for which K L-inequality fails. ◮ Note that the gradient trajectories have uniform finite length.

Kurdyka-

• Lojasiewicz

inequality and subgradient trajectories: the convex case Olivier Ley Journ´ ees Franco- Chiliennes Toulon, April 2008

An auxiliary problem

A farmer rakes its (convex) field in several steps in the following way :

ℓ0 ℓ1 ℓ2

If he is unlucky, is it possible that he walks an infinite path ? (i.e.

• i≥0

ℓi = +∞)

Kurdyka-

• Lojasiewicz

inequality and subgradient trajectories: the convex case Olivier Ley Journ´ ees Franco- Chiliennes Toulon, April 2008

An auxiliary problem

A farmer rakes its (convex) field in several steps in the following way :

ℓ0 ℓ1 ℓ2

If he is unlucky, is it possible that he walks an infinite path ? (i.e.

• i≥0

ℓi = +∞) Answer : Yes !

Kurdyka-

• Lojasiewicz

inequality and subgradient trajectories: the convex case Olivier Ley Journ´ ees Franco- Chiliennes Toulon, April 2008

Hausdorff distance between nested convex sets

• Lemma. There exists a decreasing sequence of compact convex

subsets {Tk}k in R2 such that : (i) T0 is the disk D := D(0, 2) ; (ii) Tk+1 ⊂ int Tk for every k ∈ N ; (iii)

• k∈N

Tk is the unit disk D(0, 1) ; (iv)

+∞

• k=0

distHausdorff (Tk, Tk+1) = +∞.

Kurdyka-

• Lojasiewicz

inequality and subgradient trajectories: the convex case Olivier Ley Journ´ ees Franco- Chiliennes Toulon, April 2008

Picture of the sequence of convex sets

ℓk := distHausdorff (Tk, Tk+1) ≈ Ri − Ri+1 and Ni ≈

1

√

Ri−Ri+1

It suffices to take Ri − Ri+1 = 1

i2 in order that

• i Ri − Ri+1 < ∞ and
• k ℓk ≈

i Ni(Ri − Ri+1) ≈ i

• Ri − Ri+1 = +∞.

Kurdyka-

• Lojasiewicz

inequality and subgradient trajectories: the convex case Olivier Ley Journ´ ees Franco- Chiliennes Toulon, April 2008

End of the construction

◮ Construction of a convex function with prescribed sublevel sets Tk : [Torralba 1996]. ◮ Smoothing of the function.